91Ó°ÊÓ

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{j}} \mathbf{u} $$

Find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1}\). \(\mathbf{r}(t)=\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}+\frac{1}{3} t^{3} \mathbf{k} ; t_{1}=2\)

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t^{3 / 2}, y=t^{3 / 2}, z=t ; 2 \leq t \leq 4 $$

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{i}} \mathbf{u} $$

In Problems 17-30, make the required change in the given equation. \(r=2 \sin \theta\) to Cartesian coordinates

Consider the curve \(\mathbf{r}(t)=\sin t \cos t \mathbf{i}+\sin ^{2} t \mathbf{j}+\cos t \mathbf{k}\), \(0 \leq t \leq 2 \pi\). (a) Show that the curve lies on a sphere centered at the origin. (b) Where does the tangent line at \(t=\pi / 6\) intersect the \(x y\)-plane?

Find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1}\). \(x=\sin 3 t, y=\cos 3 t, z=t, t_{1}=\pi / 9\)

Prove that the midpoints of the four sides of an arbitrary quadrilateral are the vertices of a parallelogram.

Find the equation of the surface that results when the curve \(z=2 y\) in the \(y z\)-plane is revolved about the \(z\)-axis.

Consider the curve \(\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}+\left(1-t^{2}\right) \mathbf{k}\) (a) Show that this curve lies on a plane and find the equation of this plane. (b) Where does the tangent line at \(t=2\) intersect the \(x y\)-plane?